I live in Houston, and have a couple of suggestions for how to get to Galveston. First, I've stayed at that Hyatta few times, and they do a good job, but it is not one of the higher end Hyatts...just to manage your expectations. The food is good, and they have a quiet little bar, which may get rowdy on Saturday; although I think this is primarily a business traveler's hotel. If I were you, I'd ask the hotel if they work with any shuttle companies that go to Galveston. Also, I believe the airport has a shuttle that goes to Galveston...hotel should be able to tell you that, too. If you would prefer, I use a Town Car driving service in Houston, reliable and more comfortable: Sid's Transportation, 832.287.7620. Call Sid and he'll give you pricing, etc.
I would make a taxi your LAST choice....
Also, if you have time for a leisurely, wonderful dinner that night, I highly recommend CK's at the Marriott hotel right down the street from the Hyatt, right next to the airport. CK's has a high-end menu, delicious food, wonderful service....would really set the stage for your week ahead! Just a suggestion. Enjoy your short time in Houston! txtraveler